Amir Niroomand

Analysis of Interior Point Optimization Methods for a Class of Ill-Posed Inverse Problems.

Date : November 5, 2002 (Tuesday)
Time : 3:00 - 4:00PM
Room 442 Dana Bldg

Ill-posed inverse problems require the imposition of additional constraints, beyond fidelity to measurements and a model of the physics relating the measurements to the quantities of interest, in order to stabilize against reconstruction error. One method which is sometimes employed to solve such problems is to treat all constraints, perhaps including the data and model, as sets in a solution space and to search for a solution which meets all such constraints. This is known as the admissible or feasible solution method. When these constraints are restricted to be convex, convex optimization algorithms can be applied. However these algorithms trade off between flexibility and ease of application against computational efficiency.

Diffuse Optical Tomography (DOT) is an imaging modality which requires solution of an ill-posed inverse problem. DOT attempts to reconstruct the spatial distribution of the optical absorption coefficients in biological tissue from intensity measurement on the surface of the body. This problem is typically ill-posed due to the large attenuation and scattering of the diffuse wave. In this work we use DOT as our example of an ill-posed inverse problem. We first present the Ellipsoid Algorithm as a conventional convex optimization solver for the DOT inverse problem framed using the admissible solution approach. An introduction to optimization techniquesis presented next, with emphasis on the linear matrix inequality (LMI) formalism for the DOT constrained inverse problem.

Using the SeDuMi optimizer for LMI problems, it is shown that this method has more than two orders of magnitude faster reconstruction compared to the Ellipsoid Algorithm method. It is also shown through simulation that this method results in a good reconstruction.

Committee Members:
- Professor Shafai (Advisor)
- Professor Brooks (Co-Advisor)
- Professor Miller (Committee Member)